Lemma 2 [1]} 
For fixed \(\beta >0\)  and, \(\eta (x, z)\)  defined in (REF ),
the gradient of \(\eta (x,z)\)  w.r.t. \((x, z)\)  is Lipschitz continuous with the constant \(L_{\nabla \eta }=2 \sqrt{(1+\beta )^{2}+\left(1+\frac{1}{2 \beta }\right)^{2}}\) .

